In this section, we first briefly recall the definition of (families of) projective Deligne-Mumford stacks and of the modified Hilbert polynomial and Hilbert homomorphism of sheaves. See [[1]}, [2]}, [3]}] for more details. We next recall definitions and properties of stacks (families) of expanded degenerations and pairs on smooth Deligne-Mumford stacks in [[4]}], which are important for obtaining degeneration formulas. Then we recall the definition of admissible sheaves and the associated numerical criterion, which is useful for showing the properness of good degenerations in Section 5.4.
